HPS 0410  Einstein for Everyone 
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John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
Background reading: J. P. Mc Evoy and O. Zarate, Introducing Stephen Hawking. Totem. pp. 46105,
None of the universes discussed so far are ours. To determine which universe in Einstein's great book is our universe we need to know a little more about ours. Two facts have proven decisive in selecting our universe: the distribution of matter in the universe and its motion.
How is matter distributed in our universe on the largest scale? To answer we need to get a sense of just what that largest scale is. Let us step up to it:
Within our solar system, the distance from the sun to the earth is 93 million miles; light requires 8.3 minutes to propagate from the sun to the earth. Pluto is much farther away from the sun, 2700 to 4500 million miles depending on the position in its orbit.
Our solar system is just one of hundreds of billions of stars that form our galaxy, the Milky Way. It is vastly bigger than our solar system. Its main disk is 80,000 to 100,000 light years in diameter. It is worth pausing to imagine what that means. A light year is the distance light travels in one year: 5,880,000,000,000 miles. Just one light year is already enormous. If we decide to send a light signal to some randomly chosen star in the Milky Way, it will require many tens of thousands of years to get there. That is already longer than recorded history. If some being there decides to send a signal in response, will there be anyone here to receive it?
Here's how the Milky Way looks to us from the inside as a broad luminous band made up of many stars spread across the sky.
Here's an artist's conception of how it looks from the outside:
The remaining stars of the universe are grouped into other galaxies. Here's our nearest galaxy, the Andromeda galaxy, M31, which is about 2 million light years away:
Finally, on the largest scale, luminous matter is roughly uniformly distributed through space in galaxies separated by millions of light years. Here's an image from the Hubble telescope:
The images above were drawn from the NASA website, http://www.nasa.gov/, January 14, 2007. NASA provides these images copyright free subject to the restrictions on http://www.simlabs.arc.nasa.gov/copyright_info/copyright.html
This familiar picture of the universe on the largest scale is a quite recent discovery. As late as 1920, it remained unclear whether all the matter of the universe was collected in one place, the Milky Way; or whether the Milky Way was just one galaxy of many scattered through space. This question was the subject of what came to be known as the "Great Debate" that happened in the Baird auditorium of the Smithsonian Museum of Natural History on April 26, 1920 . There two astronomers battled. Harlow Shapley defended the theory of island universe and Heber Curtis argued for the many galaxies of stars view that ultimately prevailed.
Harlow Shapey 
Heber Curtis 
These galaxies are the basic units of matter of modern cosmology. They are the molecules of the cosmic gas that is the subject of modern cosmology. The theory proceeds by assuming that they form a continuous fluid, much as we routinely assume that water or air is a continuous fluid, even though we know it is made of molecules; or that sand dunes are continuous, even though they are made of grains of sand. As long as we take a distant enough view of galaxies, molecules or sand grains, they blend into their neighbors and appear to form a continuous distribution of matter.
The galaxies form the luminous part of the matter of the universe. Recent investigations are showing that there is a lot more matter in the cosmos. It is prefixed by "dark...". Dark energy permeates all space and plays a major role in cosmic dynamics. Dark matter provides the additional gravitational pull needed to hold galaxies together.
Einstein, in 1917, presumed that on the largest scale we would see a uniform distribution of stars all roughly at rest. In the course of the 1920s, in the aftermath of the Great Debate, it became clear that the basic unit of cosmic matter would be the galaxy and not the star. That by itself changed little at the fundamental level of theory. What did change our cosmic theorizing a lot was an observation about light from distant galaxies pursued most famously by Edwin Hubble towards the end of the 1920s. That observation became the single most important observational fact of modern cosmology.
Edwin Hubble 
Mount Wilson telescope 
What Hubble observed was that light from distant galaxies was redder than light from nearby galaxies.
More importantly, there was a linear relationship between the distance to the galaxy and the amount of reddening. Double the distance and you double the reddening; triple the distance and you triple the reddening; and so on.
How was this reddening to be interpreted? Hubble inferred that it revealed a velocity of recession of the galaxies. The redder the light the faster the galaxies were receding.
Hubble arrived at this interpretation through an effect familiar from optics and acoustics, the Doppler effect. Every sound or light wave has a particular frequency and wavelength. In sound, they determine the pitch; in light they determine the color. Here's a light wave and an observer.
If the observer were to hurry towards the source of the light, the observer would now pass wavecrests more frequently than the resting observer.
That would mean that moving observer would find the frequency of the light to have increased (and correspondingly for the wavelengththe distance between creststo have decreased). That increase in frequency is a shifting of the light towards the blue end of the spectrum.
The converse effect would happen if the observer were to recede from the light source. The light's frequency would diminish and the light would redden.
For light, this effect depends only on the relative motion of observer and source. So if the observer were at rest and the light source moved, exactly the same thing would happen.  This is no longer true in the case of sound. Then there is a medium that carries the sound waves, the air, and we get slightly different results according to which of the observer or sound emitter is moving with respect to the air. There is nothing analogous to the air for lightthere is no luminiferous ether! 
The Doppler effect is familiar from everyday life. When an ambulance approaches us with its siren on, we hear a higher pitch because it is approaching. As it passes and then recedes, we hear the pitch suddenly drop. There has been no change in the sound emitted by the siren. The ambulance driver hears no change in the siren pitch. All these changes happen as a result of the relative motion between you and the ambulance siren by means of the Doppler effect.
Hubble inferred from the red shift of light from distance galaxies to a velocity of recession of the galaxies. The further a galaxies is from us, the faster it recedes. The relationship is linear, a fact to be explored in a moment.
Hubble arrived at the basic fact that all modern cosmologies try to accommodate: the universe is undergoing a massive expansion.
I'll mention here for later reference that the use of Doppler's principle as a way of interpreting the red shift has limited application. When we have developed a full cosmological model using general relativity, we'll see that the presumptions above of a static space with observers and galaxies moving in it will fail. Instead we shall see that the reddening of light from distant galaxies comes from a stretching of space itself while the light propagates to us. Doppler's principle provides a useful, classical approximation of the effect.
Hubble found a linear relationship between the velocity of recession and the distance to the galaxy. What that means can be seen in the table:
Distance to galaxy (light years) 
Velocity of recession (kilometers/second) 
1,000,000  20 
2,000,000  40 
3,000,000  60 
4,000,000  80 
5,000,000  100 
There is an obvious rule built into this table and it is known as "Hubble's Law":
Velocity of recession (kilometers/second) 
=  20  x  Distance to galaxy (millions of light years) 
The magic number of 20 in this formula carries a lot of the content. In effect is it telling us that we need to assign 20 kilometers per second of velocity of recession for every million light years of distance between us and the galaxy. This number, which is one of the most important cosmic parameters, is known as Hubble's constant.
Built into Hubble's law is also a notion of the age of the universe. To see it, consider a galaxy a million light years distant from us. If its speed of recession was the same in all history, we can compute how long ago the matter of that galaxies was here. Similarly we can compute how long ago the matter of a galaxy two million light years distant was here. And we can compute how long ago the matter of a galaxy three million light years distant was here.
A remarkable fact follows from the linearity of Hubble's law. All the times computed will come out to be the same. They will simply be one divided by Hubble's constant (with the units appropriately adjusted). The time we have computed is a time at which all the matter of the universe was coincident. That marks the beginning of the universewe now call it the "big bang." This is very pretty. We proceed from observations about galaxies to Hubble's law with its constant on to the age of the universe.
Age of Universe  =  1  /  Hubble's constant 
The Hubble age of the universe is roughly 14 billion years.
Which solutions of Einstein's gravitational field equations can accommodate a universe with a Hubble expansion? Or, more figuratively, which pages in Einstein's great book might belong to our universe?
The answer lies in a class of solutions of Einstein's equations picked out a few simple conditions, FriedmannRobertsonWalker spacetimes. These spacetime can be sliced up into spaces that evolve into each other over time.
(It isn't automatic that a spacetime can be cut up into nice spatial slices. A Goedel universe cannot be sliced up nicely into spaces that evolve into each other over time.)
What character should the spaces have? Our observations of our cosmos tell us that on the largest scale space is homogeneous and isotropic. So we ask that the solutions have a homogeneous, isotropic spacethat is a space that is the same in every place (homogeneous) and in all directions (isotropic); and that this space simply evolves with time.
This condition that space is homogeneous and isotropic on the largest scale has been called the "cosmological principle." That strikes me as a little dangerous. Naming the condition the cosmological principle does no harm, of course, as long as we realize that it is just a name. However, when the term "principle" is used, it is easy to get the impression that the condition is somehow unchallengeable. That is risky. Whether the universe is roughly homogeneous and isotropic is something that should be determined by observation. It should not be elevated to apriori heights.
The condition that the space is homogeneous and isotropic restricts it to three general possibilities. Such a space must have constant spatial curvature. We know from earlier that the three possibilities are:
Spherical Positive curvature 
Flat, Euclidean Zero Curvature 
Hyperbolic Negative Curvature 
A space of one of these three types will be the instantaneous snapshots that comprise the "now" of the cosmology.
Each of these snapshots of space will be filled with a uniform matter distribution. Its composition is not fixed. It may be ordinary matter, such as comprise planets and stars; or it may be radiation; or it may be a mixture of the two. At present we have a mixture that is heavily skewed towards ordinary matter. In the past, radiation was dominant.
Finally the spaces of the cosmology cannot remain static.
They are either expanding or contracting. The first
case of expansion is the one that interests us most since it is what we
observe. As time passes, the space expands, its curvature, if it has any
decreases, and the distance between the galaxies increases. The figure shows
the worldlines of the galaxies with the spatial slices.
The most interesting feature of this figure is what happens when we project the worldlines of the galaxies into the past. The galaxies get closer and closer. Eventually, they converge onto a state of infinite curvature and density. This is the initial statethe socalled "big bang."
It is easy to misunderstand the nature of the big bang and the expansion of the universe.
The popular image called to mind by the name big bang is something like this. There is a huge empty space, with an infinitely dense nugget of matter containing all future matter of the universe. At the moment of the big bang, This nugget explodes. Fragments of this primeval nugget are scattered into space, progressively filling it with an expanding cloud of matter. This is NOT the modern big bang model.
Rather the expansion is the expansion of space itself. The most helpful picture is of the rubber surface of a balloon expanding. The galaxies are like dots drawn on the surface. They move as the rubber sheet stretches. The galaxies fly apart because space expands. At any instant, space is always full of matter; there is no island of fragments expanding into an empty space.
If we now project back to the big bang, we project back to a time at which all matter and space were somehow compressed into a state of infinite density. Einstein's gravitational field equations tell us the matter density equals the summed spacetime curvature. So, if the matter density is infinite, the curvature of spacetime has become infinite as well.
That last statement cannot be literally correct. According to Einstein's general theory of relativity, spacetime at every event has definite curvature. If that curvature is everywhere infinite, we define no spacetime at all. If we try to imagine the time of the big bang itself as one of the times of the cosmology, we are saying that there is a time at which spacetime is not properly defined. So there can be no time in the cosmology corresponding to the big bang. We describe the big bang as a "singularity," a breakdown in the laws that govern space and time.
The term
singularity, roughly speaking, designates a point in a mathematical
structure where a quantity fails to be well defined, even though the quantity
is well defined at all neighboring points. The simplest and best known example
arises with the inverse function, 1/x. As long as x is nonzero, 1/x is well
defined. For
x = 10, 5, 1, 0.5, 0.1, 0.01, ...,
1/x = 0.1, 0.2, 1, 2, 10, 100, ...
For negative values
x = 10, 5, 1, 0.5, 0.1, 0.01...,
1/x is 0.1, 0.2, 1, 2, 10, 100, ...
The system has a singularity at x=0, for then 1/x = 1/0 and, as we
all learn in our arithmetic classes, "you cannot divide by zero." There is a
temptation to say that 1/0 is "infinity." But that is dangerous. As we have
just seen, if we approach x=0 from positive values of x, the inverse 1/x grows
without limit towards +infinity (i.e. "plus infinity"). If we approach x=0 from
negative values of x, the inverse 1/x grows negatively without limit towards
infinity (i.e. "minus infinity"). If we insist on giving 1/0 a value, which do
we give? "Plusinfinity" or "minusinfinity"? The safer course is just to say
that we have a singularity at x=0 and not try to give it any value.
What we can say is this. The universe has an age or timeits age after the big bang. The spacetime of the universe exists for every age greater than zero: 1 million years, one hundred years, one second, one half second, one tenth second, and so on. No matter how small we make the age, there is a corresponding spacetime, as long as the age is greater than zero. But nothing corresponds to the zero age.
This moment of zero age is a fictitious moment in the history of the universe. In that regard, it is like the fictitious point "at infinity" on the horizon where parallel lines meet. Of course well all know that there is no such point, although we see it drawn routinely in perspective drawings.
We can now return to the red shift that figures in the Hubble expansion and give a more precise account of its origin. It is not a traditional Doppler shift, but something more subtle. A distant galaxy emits light towards us. The light waves with their crests are carried by space towards us. For a distant galaxy, it can take a very long time for the light to reach us. During that time, the cosmic expansion of space proceeds. The effect is that the waves of the light signal get stretched with space. So the wavelength of the light increases and its frequency decreases. It becomes red shifted.
To get a sense of the process, imagine a column of ants setting off to walk across a rubber sheet. They may enter the sheet at a rate of one ant per second. If the rubber sheet is stretched while the ants walk, each ant will need to go further to get to the other side than the one before. So the ants will arrive less frequently at the other side than the original rate of one ant per second.
What is the overall dynamics of spacetime? Einstein's gravitational field equations applied to the FriedmannRobertsonWalker spacetimes give us three possibilities, cataloged below as I, II or III.
What decides between them is the density of matter. The
socalled "critical density" of matter is the
deciding value. It is a minute average density of 10^{29}grams per
cubic centimeter. Our cosmology will be one of the three shown in the table
below according to whether the actual average density of matter in our universe
is greater than, equal to or less than this critical density.
Cosmology  I  II  III 

Average mass density  Greater than critical  Critical  Less than critical 
Geometry of space  Spherical positive curvature 
Flat, Euclidean zero curvature 
Hyperbolic negative curvature 
Dynamics  Expands and collapses to big crunch  Expands indefinitely  Expands indefinitely 
The table gives the broad features. In cases I and III, space is curved. The
scale factor Rthe radius of curvature of the
spacedetermines the extent of curvature. (The radius of curvature of a three
dimensional space is the threedimensional analog of the radius of a
twodimensional sphere.) The value of R differs greatly according to the
particular matter density at hand. However a rough estimate is this:
Scale factor R 
very roughly equals 
Hubble age of universe 
x  speed of light 
So by this estimate the scale factor is roughly 14 billion light years. This value only obtains exactly for special cases. In cosmologies I, it obtains exactly if the average matter density is twice the critical.
We can also get a sense of the dynamics by plotting how the scale factor R changes with time in typical examples of the three cosmologies. In the case of cosmologies II with Euclidean geometry, the scale factor R is simply set to be the distance between two conveniently placed galaxies. As the cosmic expansion proceeds, R grows in response.
In general there are no simple formulae for these curves. One case proves to be simple. In Cosmologies II, if all the matter is what is called "dust" in the jargon (i.e. ordinary matter like our earth), then R increases in direct proportion with (time)^{2/3}. Or in that cosmology, if all the matter is radiation, R increases in direct proportion with (time)^{1/2}.
At first the dynamics seems arbitrary. Why should the different universes have the properties they do? Why, for example, should a universe with greater mass density only have a big crunch? And why with lesser mass density, will the expansion continue indefinitely? We can makes some sense of this with an analogy from Newtonian theory.
There is a reason Newtonian theory can tell us something. Recall that general relativity turns back into Newtonian theory as long as we consider ordinary conditions: nothing moves quickly, there are no strong gravitational fields andmost important herewe consider small distances, not cosmic distances.
So it turns out that a tiny chunk of the cosmic fluid of a FriedmannRobertsonWalker spacetime is governed by Newtonian principles. The easiest way to see those principles in action is to consider a closely analogous system in Newtonian theory.
Imagine that we have a bomb in space that explodes. It will spread debris into space. Each fragment in the debris will attract all the others according to Newton's inverse square law of gravity. The ultimate fate of the debris cloud depends on the balancing of the initial magnitude of the explosion with the strength of the gravitational attraction within the debris cloud.
If there is a greater amount of matter in the original lump, the explosion will produce a denser cloud of debris. Its internal forces of gravitational attraction will be strong enough to slow and halt the initial outward motion of the explosion and draw the fragments back together, bringing about a collapse. It corresponds to the dynamics of cosmology I; there is a big bang and a big crunch.
If there is a lesser amount of matter in the original lump, the explosion will produce a more dilute debris cloud whose internal forces of attraction will not be sufficient to halt the initial outward motion of the blast. That outward motion will continue indefinitely. It corresponds to cosmology III; there is a big bang and no big crunch.
We could imagine an intermediate case in which the explosion is just energetic enough to fling the debris out of the reach of the gravitational forces; any weakening of the explosion would be too weak to prevent collapse. This corresponds to the intermediate case of cosmology II.
The Newtonian analogy is useful in so far as it gives us a nice picture for the dynamics. But it omits a lot. There is no account of the different spatial geometries and the big bang is the explosion of a nugget of matter into a preexisting space. That is not what is portrayed by relativistic cosmologies.
Which of these three cosmologies is ours? The questions is not without some interest. If it is the first cosmology I, then we live in a finite space. If we point in in any direction, after some finite distance, we are pointing at the back of our own heads! Further, just as the universe has a finite past bounded by the big bang, so there is also an end in our future. The entire universe will collapse down onto itself in a "big crunch". In cosmologies II and III neither of these results obtain. Cosmology II, however, is the only one in which the geometry of space on cosmic scales is Euclidean.
The value of the critical density is extremely small: 10^{29}grams per cubic centimeter of space. That is 0.00000000000000000000000000001 grams per cubic centimeter. That is very little indeed! It corresponds to roughly 5 hydrogen atoms only in a cubic meter of space. That sort of vacuum is extremely hard to achieve with laboratory equipment on earth.
Here's another measure of how small it is. Take
one fifth of a teaspoon of water, which is
roughly 20 drops. (That amounts to one gram.) How widely spread must it
be in order to match the critical density? Guess! What if we take those
20 drops and spread them over the volume of the Astrodome? Not even
close. Think bigger. What about 20 drops spread over the volume of
earth? Better, but still too small. That density is still 100 times too
big. Those 20 drops of water need to spread over one hundred earth
volumes if their dilution is to match the critical density! That, at least, is what my sums show. The radius of the earth is about 6,366,000 meters. So its volume is 1.08 x 10^{21} m^{3}, which comes to 1.08 x 10^{27} cubic centimeters. So one gram spread over this volume is still roughly 100 times too dense. 
Since this critical density is so small, you might think that our universe must have an average density more than critical. That would be jumping to conclusions. What counts is the density of matter averaged over all space. So we need to take the matter of earth and spread it over the vast emptiness of space between stars and galaxies. And then the calculation gets more complicated because of the steady accumulation of evidence that a very substantial portion of the energy of our universe is "dark," so its existence is actually inferred indirectly from the gravitational effects it produces.
The upshot is that the average density of matter comes out very close to the critical. Indeed the astonishing and maddening result is that the more accurately it is measured, the closer our density gets to the critical value. So we remain unable to say which of the cosmic scenarios above is our own.
The suspicion is growing that our density may be exactly the critical density. It seems too much of a coincidence that of all values our matter density could have, it just turns out to be so close to the critical density. So the supposition is that there might be some cosmic process that has driven the matter density to this value. So called "inflationary" cosmologies posit an early phase of very rapid cosmic expansion that would have the effect of driving the matter density towards the critical.
The distinctive feature of big bang cosmology is the big bang. We know it is there because when we project back the trajectories of the expanding galaxies and see that they all converge onto one point. It is somewhat like a lens focussing the rays of light of the sun. The rays emerge from the lens just perfectly aligned to focus to a single infinitely bright spot.  http://en.wikipedia.org/wiki/File:Lens1.svg 
http://en.wikipedia.org/wiki/File:Large_convex_lens.jpg 
Or that is what they would do in ideal circumstances. That is, if the light rays falling onto the lens were perfectly parallel and the lens perfectly constructed. In the real world, there are always slight unevennesses and neither of these assumptions hold. All that will happen is that the light is focussed to a very bright spot, not a point of infinite intensity. 
Might the same be true of the big bang? FriedmannRobertsonWalker spacetimes presume a perfectly uniform matter distribution. That means that all the motions of matter now are presumed symmetrically arranged so that, from the uniformity alone, if we project back into the past, we eventually come to a singular point. 
That perfect uniformity is an idealization we know
is not true of our world. While the matter of the universe might be
nearly uniform when seen on some cosmic scale, locally it is far from
uniform. When we allow for these nonuniformities might we not have a
big banga true singularitybut merely region of spacetime with a lot
of near misses? So instead of the big bang, we have a temporary region of very high density? If that
happened, the big bang would no longer be the beginning of time. There
would be time and matter and space before the big bang. The big bang
would merely be an extremely hot phase of highly compresses matter and
space in the overall history of the cosmos.
Might the big bang merely be an unrealistic artefact of an unrealistic symmetry assumption? For the big bang to be physically interesting to us, its existence must be assured robustly by physical principles, not fragile assumptions of uniformity. 
It was long supposed that nonuniformities might preclude a true singularity. In the 1960s, when a group of mathematical physicists turned to this and related issues, it was soon shown that this supposition was wrong.There was a great deal more inevitability to the big bang. The results were delivered in the form of mathematical theorems within the framework of general relativity. They show that, under quite broad conditions, even with nonuniformity, a singularity is inevitable. Here is one of a number of these theorems proved by Stephen Hawking.
IF  
(a) the universe is expanding at some instant
("now"); 

(b) the rate of expansion is now everywhere greater
than some fixed positive amount "K"; 
Note that this expansion need not be uniform. It can be high in one place and low in another. It just must be everywhere greater than some positive amount K. This "K" can be anything, but it must be greater than zero. 
(c) Einstein's gravitational field equations (without λ) hold with the "strong energy condition";  Recall that the notion of "matter density" in general relativity is
complicated. Energy contributes to it, but so do stresses. Indeed
stresses can be sources of the gravitational field. The "strong energy
condition" requires that the contribution to matter from ordinary
energy is greater than that from stresses. So this condition says "not
too much funny matter."
Notice that there is no condition that the matter distribution be uniform or even that there be matter everywhere. 
(d) there are no causally isolated pockets of spacetime in the universe;  The technical requirement is that the spatial slice "now" be a Cauchy surface. A Cauchy surface has the property that every nonterminating physical process, propagating at or less than the speed of light, must pass through it exactly once. So any process in the surface's past will end up intersecting it; and any process in its future must have passed through it. Specifying the state of a Cauchy surface fixes the future and past geometric state of the spacetime fully. That a spacetime have a Cauchy surface is a strong condition not satisfied by many spacetimes. The hypersurfaces of simultaneity of a Minkowski spacetime and the natural spatial slices of a FriedmannRobertsonWalker spacetime representing the "nows" are Cauchy surfaces. 
THEN 

no timelike world line can be extended indefinitely into the past; that is, no timelike curve has greater temporal length than 3/K. 
The "THEN" conclusion in effect tells us that there is something pathological going on in our past. If we try to trace the history of a galaxyone case of a timelike world lineindefinitely into the past, something blocks it. Such world lines cannot extend back arbitrarily. You might complain that this is an oblique way of characterizing a singularity. That is true, but that is how these matters are dealt with. Recall that a singularity is not a point in the spacetime, so its identification will have to be indirect.
We can get a more intuitive sense of how the theorems work by recalling the jig saw puzzle analogy for solving Einstein's equations. We specify the "now" part of spacetime in accord with the "IF" conditions above. We then try to reconstruct the spacetime of the past by solving Einstein's gravitational field equations; that is, we start to put in the pieces of spacetime that fill out the past. What we discover if that we can only keep adding in pieces for some finite distance in time to the past. Then we can go no further.
The strength of a theorem is that it is a mathematically proven result. As long as the "IF" conditions are met, the "THEN" conclusion is forced by mathematics alone. That is also the weakness. The "IF" conditions may fail. Indeed stating a theorem like this is an invitation to troublemakers to find failures of the IF condition.
They certainly can be found. If there are black holes, then the causal niceness condition (d) is violated. Also, as we discover more and more exotic forms of matter in the cosmos, we may worry about the energy condition (c). Or, if Einstein's cosmological constant λ is small but nonzero in a world with spherical spatial geometry, then it turns out that we can have a cosmology with no big bang and no big crunch. Space slowly collapses over time to some minimum size and then expands out again. It is a single gentle bounce.
Copyright John D. Norton. March 2001; January 2007, February 16, 23, October 16, November 10, 2008, March 31, 2010.